Which of the following methods uses Euler’s method as a predictor step with a full step size?

Question

BlackTom AI · Accepted Answer

Heun’s method is the explicit predictor–corrector pair that uses an Euler forward step as the predictor with the full step size h, so the correct answer is b. Heun.

1) Relevant knowledge points and concepts
- Euler (explicit) method: single-step first-order method y_{n+1} = y_n + h f(t_n,y_n). No corrector stage.
- Predictor–corrector methods: generate a provisional value (predictor) and then refine it (corrector), often averaging slopes to raise accuracy.
- Heun’s method (also called the improved Euler or explicit trapezoidal rule): uses Euler’s full-step prediction and then corrects by averaging the slope at the beginning and at the predicted end. It is a second-order method.
- Midpoint method: a two-stage Runge–Kutta that uses a half-step prediction to evaluate the slope at the midpoint, not a full-step Euler predictor.

2) Step-by-step reasoning with formulae
- Euler predictor (full step): y_pred = y_n + h f(t_n,y_n).
- Heun corrector (average slopes): y_{n+1} = y_n + (h/2)[ f(t_n,y_n) + f(t_{n+1}, y_pred) ].
These two lines show explicitly that Heun’s predictor is exactly the Euler forward formula with step size h, and the corrector averages the initial slope and the slope evaluated at the Euler-predicted endpoint.

3) Intermediate steps (symbolic demonstration)
- Compute k1 = f(t_n,y_n).
- Predictor: y_pred = y_n + h k1.
- Compute k2 = f(t_{n+1}, y_pred).
- Corrector: y_{n+1} = y_n + (h/2)(k1 + k2).

4) Why this is correct
- The predictor used in Heun is exactly Euler’s method with full h (y_pred = y_n + h f). That distinguishes Heun from midpoint, which evaluates at t_n + h/2 and uses a half-step predictor y_mid = y_n + (h/2)f(t_n,y_n).

5) Why other options are incorrect
- a. Euler: Euler itself does not use Euler as a predictor plus a corrector — it has no correction stage.
- c. Midpoint: uses a half-step predictor (t_n + h/2) and evaluates f at the midpoint; not Euler with full h.
- d. None of the above: incorrect because Heun explicitly matches the description.

Conclusion: This reasoning confirms that option b. Heun is the correct choice, matching the provided answer.

Which of the following methods uses Euler’s method as a predictor step with a full step size?单项选择题

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